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ISyE 3133 C  Engineering Optimization
Homework Assignment # 2
Issued:
August 26, 2009
Due Date:
September 4, 2009
1.
[15 points] Answer the following questions:
a)
What are the definitions of linear independence, subspace, linear span of a set of vectors, and
basis of a subspace?
b) Consider the vectors v
1
, .
.., v
k
Ը
n
. Recall that v
1
Ը
n
means that the vector v
1
has n
elements, each of which is a real number. Suppose v
1
, .
.., v
k
are linearly independent.

Can k > n? Why?

Is
Ը
n
a subspace? Why?

What would k have to equal for v
1
, .
.., v
k
to be a basis for
Ը
n
?
c) Suppose
A
Ը
3 x 4
. Recall that this means that
A
is a matrix with 3 rows and 4 columns and all
its entries are real numbers.

What is the maximum possible number of linearly independent columns of
A
?

What is the definition of
rank(
A
)
?

What can you say about
rank(
A
)
from the fact that
A
Ը
3 x 4
?
d)
Given a nonsingular square matrix
B
.

What is the definition of its inverse
B
−
1
?

What can you say about
det(
B
)
 the determinant of
B
?
e)
Given a matrix
B
Ը
mxm
with inverse
B
−
1
, a vector b
Ը
m
, and the system of equations
B
x = b,
solve for x
Ը
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This note was uploaded on 09/07/2009 for the course COE 2001 taught by Professor Valle during the Fall '08 term at Georgia Institute of Technology.
 Fall '08
 VALLE

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